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September 30th, 2014, 07:53 AM   #1
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How to prove that $2$ is an irreducible element in $\mathbb {Z}[\sqrt{5}]$?

How to prove that $2$ is an irreducible element in $\mathbb {Z}[\sqrt{5}]$? One way is to check if there exists $a,b,c,d \in \mathbb Z$ such that $2=(a+b\sqrt{5})(c+d\sqrt{5})$, i.e., $2=ac+ 5bd+ \sqrt{5}(ad+bc)$. It leads to the system $ac+5bd=2$ and $ad+bc=0$, which is difficult to solve. Is there another way? Thanks!
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October 4th, 2014, 03:08 AM   #2
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For $a + b\sqrt{5} \in \Bbb Z[\sqrt{5}]$ define its norm $N$ by:

$N(a + b\sqrt{5}) = a^2 - 5b^2$.

Note that:

$N((a + b\sqrt{5})(c + d\sqrt{5})) = N((ac + 5bd) + (ad + bc)\sqrt{5}) = (ac + 5bd)^2 - 5(ad + bc)^2$

$=a^2c^2 + 10abcd + 25b^2d^2 - 5a^2d^2 - 10abcd - 5b^2c^2$

$= a^2c^2 - 5b^2c^2 - 5a^2d^2 + 25b^2d^2 = (a^2 - 5b^2)(c^2 - 5d^2) = N(a + b\sqrt{5})N(c + d\sqrt{5})$.

So $N$ is multiplicative.

Clearly, if $a + b\sqrt{5}$ is a unit, then, we must have $N(a + b\sqrt{5}) = \pm 1$.

Suppose, for the sake of argument, $N(a + b\sqrt{5}) = -1$, so:

$1 = 5b^2 - a^2$. If $|a| \geq |b|$, then:

$1 = 5b^2 - a^2 \geq 4b^2$. This is a contradiction, unless $b = 0$, in which case we have: $1 = -a^2$, which is impossible.

Similarly, if $|a| < |b|$, then:

$1 = 5b^2 - a^2 > 4a^2$ which forces $a = 0$, leading to a similar contradiction.

Thus if $a + b\sqrt{5}$ is a unit, it has norm 1. The converse is clear: if $N(a + b\sqrt{5}) = 1$, then $a^2 - 5b^2 = 1$, in which case:

$a + b\sqrt{5}$ has inverse $a - b\sqrt{5}$. For example, $9 + 4\sqrt{5}$ is a unit.

Now if $2 = (a + b\sqrt{5})(c + d\sqrt{5})$, then:

$4 = (a^2 - 5b^2)(c^2 - 5d^2)$.

As we saw above, NO element of $\Bbb Z[\sqrt{5}]$ has norm -1, and if one of the factors of 2 has norm 1, it is a unit, so it doesn't count (a ring element is reducible only if it factors into two non-units).

This means if 2 is reducible, its non-unit factors must each have norm $\pm 2$.

The same argument we used to show no element of $\Bbb Z[\sqrt{5}]$ has norm -1 also works to show that no element has norm -2.

So we must have $a^2 = 5b^2 + 2$. Here, we can use a clever trick:

$a^2 = 2$ (mod 5). Checking, we see that:

$0^2 = 0$ (mod 5)
$1^2 = 1$ (mod 5)
$2^2 = 4$ (mod 5)
$3^2 = 4$ (mod 5)
$4^2 = 1$ (mod 5), so there is no solution.

(If you prefer, write $a = 5k + n$ for $n = 0,1,2,3,4$, and put all the multiples of 5 on one side of the equation).

My apologies for the length of this post.
Thanks from Ould Youbba and walter r
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